Metamath Proof Explorer

Theorem clwlks

Description: The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018) (Revised by AV, 16-Feb-2021) (Revised by AV, 29-Oct-2021)

		Ref	Expression
	Assertion	clwlks	\|- ( ClWalks ` G ) = { <. f , p >. \| ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	\|- ( g = G -> ( ( p ` 0 ) = ( p ` ( # ` f ) ) <-> ( p ` 0 ) = ( p ` ( # ` f ) ) ) )
2		df-clwlks	\|- ClWalks = ( g e. _V \|-> { <. f , p >. \| ( f ( Walks ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )
3	1 2	fvmptopab	\|- ( ClWalks ` G ) = { <. f , p >. \| ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }

Step

Hyp

Ref

Expression

biidd

 |-  ( g = G -> ( ( p ` 0 ) = ( p ` ( # ` f ) ) <-> ( p ` 0 ) = ( p ` ( # ` f ) ) ) )

df-clwlks

 |-  ClWalks = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } )

1 2

fvmptopab

 |-  ( ClWalks ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) }